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Analytical and Finite Element Based Micromechanics for Failure Theory of Composites

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Title:
Analytical and Finite Element Based Micromechanics for Failure Theory of Composites
Creator:
Kotikalapudi, Sai Tharun
Place of Publication:
[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (92 p.)

Thesis/Dissertation Information

Degree:
Master's ( M.S.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mechanical Engineering
Mechanical and Aerospace Engineering
Committee Chair:
SANKAR,BHAVANI V
Committee Co-Chair:
KUMAR,ASHOK V

Subjects

Subjects / Keywords:
abaqus -- admm -- analytical -- composite -- dmm -- envelopes -- failure -- fea -- fiber -- halpin-tsai -- hashin -- interface -- matrix -- micromechanics -- microstress -- phenomenological -- prediction -- properties -- rve -- sankar -- strengths -- three-phase -- unidirectional -- volume-fraction -- von-mises
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Mechanical Engineering thesis, M.S.

Notes

Abstract:
An analytical method using elasticity equations to predict the failure of a unidirectional fiber-reinforced composite under multi-axial stress is presented. This technique of calculating micro-stresses using elasticity equations and estimating the strengths of a composite is based on the Direct Micro-mechanics Method (DMM). Prediction of failure using phenomenological failure criteria such as Maximum Stress, Maximum Strain and Tsai-Hill theories have been prevalent in the industry. However, DMM has not been used in practice due to its prohibitive computational effort such as the finite element analysis (FEA). The present method replaces the FEA in DMM by analytical methods, thus drastically reducing the computational effort. A micromechanical analysis of unidirectional fiber-reinforced composites performed using the three-phase model. A given state of macro-stress is applied to the composite and the micro-stresses in the fiber and matrix phases and along the fiber-matrix interface are calculated. The micros-stresses in conjunction with failure theories for the constituent phases are used to determine the integrity of the composite. The analytical model is first verified by comparing with results for finite element based micro-mechanics. Then, it is used to study the failure envelopes of various composites. The effects of fiber-matrix interface on the strength of the composite is studied. The results are compared with those available in the literature. It is found that the present analytical Direct Micro-mechanics (ADMM) predicts the strength of composites reasonably well. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (M.S.)--University of Florida, 2017.
Local:
Adviser: SANKAR,BHAVANI V.
Local:
Co-adviser: KUMAR,ASHOK V.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2018-06-30
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by Sai Tharun Kotikalapudi.

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Applicable rights reserved.
Embargo Date:
6/30/2018
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LD1780 2017 ( lcc )

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ANALYTICAL AND FINITE ELEMENT BASED MICROMECHANICS FOR FAILURE THEORY OF COMPOSITES By SAI THARUN KOTIKALAPUDI A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2017

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2017 Sai Tharun Kotikalapudi

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To Amma and Nanna for the incessant support and always believing in me

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4 ACKNOWLEDGMENTS I would like to express my gratitude to my thesis advisor Dr. Bhavani V. Sankar for being a very supporting guide and mentor, and for giving me a chance to participate in his research program critical to composite industry. He has consistently steered me in the right direction and been a lamppost on this hazy road of exploration. I w ould also like to thank Dr. Ashok V. Kumar for willing to be a member of the supervisory committee and offer constructive criticism wherever needed. I am grateful for my parents and Srinivas Namagiri for all their support and encouragement. Without them t his day would have never dawned. I am beholden to Sumit Jagtap for his guidance on Abaqus as well as to Apoorva Walke and Maleeha Babar for being a constant source of moral support I would also like to acknowledge all my teachers from University of Florid a and SASTRA University, and my friends and family for being there for me in the time of need.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST O F TABLES ................................ ................................ ................................ ............ 6 LIST OF FIGURES ................................ ................................ ................................ .......... 8 LIST OF ABBREVIATIONS ................................ ................................ ........................... 11 ABSTRACT ................................ ................................ ................................ ................... 12 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 14 Litera ture Review ................................ ................................ ................................ .... 14 Research S cope ................................ ................................ ................................ ..... 15 2 ANALYTICAL EQUATIONS ................................ ................................ .................... 20 Introduction to the Three Phase Model ................................ ................................ ... 20 Halpin Tsai Formulation for Composite Properties ................................ ................. 21 Longitudinal and Hydrostatic Stress Equations ................................ ....................... 23 Longitudinal Shear Stress in the x y plane ................................ .............................. 27 Longitudinal Shear Stress in the x z plane ................................ .............................. 32 Biaxial tension/compression in y z pla ne ................................ ................................ 37 Transverse Shear E quations ................................ ................................ .................. 43 3 FINITE ELEMENT ANALYSIS AND COMPARISON ................................ .............. 50 Modelling and analysis of Hexagonal RVE ................................ ............................. 50 Comparison with analytical model ................................ ................................ .......... 61 4 ANALYTICAL MODEL RESULTS AND DISCUSSION ................................ ........... 65 Results for Kevlar/Epoxy ................................ ................................ ......................... 65 Carbon/Epoxy plots ................................ ................................ ................................ 69 Effects of Interface ................................ ................................ ................................ .. 73 Volume fraction analysis ................................ ................................ ......................... 81 Summary ................................ ................................ ................................ ................ 84 5 CONCLUSIONS AND FUTURE WORK ................................ ................................ 87 LIST OF REFERENCES ................................ ................................ ............................... 90 BIOGRAPHICAL SKETCH ................................ ................................ ............................ 92

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6 LIST OF TABLES Table page 2 1 Comparison of macro stresses with average micro stresses for longitudinal shear stress ................................ ................................ ................................ ........ 22 2 2 Comparison of macro stresses with average micro stresses for normal Stress and in plane shear stress ................................ ................................ ................... 22 3 1 Properties of Kevlar/Epoxy used in the FEA ................................ ....................... 52 3 2 Coefficients of stiffness matrix obtained from unit strain analysis ....................... 56 3 3 Transverse strengths at various points for Kevlar/Epoxy (plane s train) .............. 60 3 4 Comparison of various transverse strengths for Kevlar/Epoxy (plan e s train) ..... 63 3 5 Comparison of maximum principal stress for Kevlar/Epoxy (plane s train) .......... 63 3 6 Comparison of maximum von Mises str ess for Kevla r/Epoxy (plane s train) ....... 63 3 7 Comparison of average of top 10% maximum principal st resses for Kevlar/Epoxy (plane s train) ................................ ................................ ................ 63 3 8 Comparison of average of top 10% von Mises st resses for Kevlar/Epoxy (plane s train) ................................ ................................ ................................ ....... 63 3 9 Comparison of 10 th percentile maximum principal stress for Kevlar/Epoxy (plane s train) ................................ ................................ ................................ ....... 64 3 10 Comparison of 10 th percentile maximum von Mises stress for Kevlar/Epoxy (plane s train) ................................ ................................ ................................ ....... 64 4 1 Properties of Kevlar/Epoxy ................................ ................................ ................. 66 4 2 Strengths at various points for Kevlar/Epoxy (MPa) ................................ ........... 69 4 3 Properties of Carbon T300/Epoxy 5208 ................................ ............................. 70 4 4 Predicted strengths of T300/5208/Carbon/Epoxy ................................ ............... 73 4 5 Comparison of strengths for Kevlar/epoxy including interface failure obtained using ADMM ................................ ................................ ................................ ....... 80 4 6 Comparison of strengths for Carbon/Epoxy including interface failure obtained using ADMM ................................ ................................ ........................ 81

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7 4 7 Comparison of strengths for several composites with analytical model strengths ................................ ................................ ................................ ............. 85 4 8 %Difference of strengths for several composites relative to reference strengths ................................ ................................ ................................ ............. 85

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8 LIST OF FIGURES Figure page 1 1 Depiction of a RVE for the analytical model ................................ ....................... 16 1 2 Decomposition of macro stresses applied to an RVE of a fiber composite ......... 16 1 3 Macro stresses applied on the unit cell. (similar to will be acting in the 13 plane a nd will be acting in the 2 3 plane ) ................................ .......... 17 1 4 Decomposition of applied state of macro stresses into five cases ...................... 18 2 1 Three phase model ................................ ................................ ............................ 20 3 1 Representative volume element of a hexagonal unit cell ................................ .... 50 3 2 Coordinate system used in ABAQUS and principal coordinate system .............. 51 3 3 Sectional view and dimensions of the RVE ................................ ........................ 51 3 4 Meshed RVE, red bounded regions represent fiber and green unbounded region represents matrix ................................ ................................ ..................... 52 3 5 Element ty p e used for meshing and analysis ................................ ..................... 53 3 6 Boundary conditions and loading in unit strain analysis (A) Direction 2 (B) direction 3 ................................ ................................ ................................ ........... 54 3 7 Schematic of the procedure foll owed to obtain Stiffness matrix .......................... 5 5 3 8 Initial and deformed hexagonal RVE under unit strain in A) 2 nd Direction B) 3 rd direction C) 2 nd and 3 rd direction ................................ ................................ ..... 56 3 9 Schematic of procedure to plot a failure envelope in 2 3 plane .......................... 59 3 10 Failure envelopes of Kevlar/Epoxy in transverse direction obtain ed through unit strain analysis ................................ ................................ .............................. 60 3 11 Comparison of analytical and finite element model failure envelopes using maximum stress theory in the transverse plane (2 3 plane) ............................... 61 3 12 Comparison of analytical and finite element model failure envelopes using quadratic theory in the transverse plane (2 3 plane) ................................ .......... 62 4 1 Comparison of MMN and QQN failure envelopes of Kevlar/Epoxy on plane ................................ ................................ ................................ ................... 66

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9 4 2 Comparison of MMN and QQN failure envelopes of Kevlar/Epoxy in the plane ................................ ................................ ................................ .............. 67 4 3 Comparison of MMN and QQN failure envelopes of Kevlar/Epoxy for longitudinal shear in the 1 2 or 1 3 plane ................................ ........................... 67 4 4 Comparison of MMN and QQN failure envelopes of Kevlar/Epoxy subjected to both longitudinal and transverse shear stresses ................................ ............. 68 4 5 Comparison of MMN and QQN failure envelopes of Kevlar/Epoxy for shear in longitudinal direction and stress in fiber direction ................................ ............... 68 4 6 Comparison of MMN and QQN failure envelopes of Carbon/Epoxy in 1 2 plane ................................ ................................ ................................ ................... 70 4 7 Comparison of MMN and QQN failure envelopes of Carbon/Epoxy in 2 3 plane ................................ ................................ ................................ ................... 71 4 8 Comparison of MMN and QQN failure envelopes of Carbon/Epoxy for shear in longitudinal directions ................................ ................................ ..................... 71 4 9 Comparison of MMN and QQN failure envelopes of Carbon/Epoxy subjected to both longitudinal and transverse shear stresses ................................ ............. 72 4 10 Comparison of MMN and QQN failure envelopes of Carbon/Epoxy for longitudinal shear and normal stress in fiber direction ................................ ........ 72 4 11 Interface effects on failure envelopes for Kevlar/Epoxy in 1 2 plane using maximum stress theory ................................ ................................ ...................... 74 4 12 Interface effects on failure envelopes for Kevlar/Epoxy in 1 2 plane using quadratic theory ................................ ................................ ................................ .. 75 4 13 Interface effects on failure envelopes for Kevlar/Epoxy in 2 3 plane using maximum stress theory ................................ ................................ ...................... 75 4 14 Interface effects on failure envelopes for Kevlar/Epoxy in 2 3 plane using quadratic theory ................................ ................................ ................................ .. 76 4 15 Interface effects on failure envelopes subjected to both longitudinal and transverse shear stresses using maximum stress theory ................................ ... 76 4 16 Interface effects on failure envelopes subjected to both longitudinal and transverse shear stresses using quadratic theory ................................ .............. 77 4 17 Interface effects on failure envelopes for Carbon/Epoxy in 1 2 plane using maximum stress theory ................................ ................................ ...................... 77

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10 4 18 Interface effects on failure envelopes for Carbon/Epoxy in 1 2 plane using quadratic theory ................................ ................................ ................................ .. 78 4 19 Interface effects on failure envelopes for Carbon/Epoxy in 2 3 plane using maximum stress theory ................................ ................................ ...................... 78 4 20 Interface effects on failure envelopes for Carbon/Epoxy in 2 3 plane using quadratic theory ................................ ................................ ................................ .. 79 4 21 Interface effects on failure envelopes for Carbon/Epoxy for envelopes subjected to both longitudinal and transverse shear stresses ............................ 79 4 22 Interface effects on failure envelopes for Carbon/Epoxy longitudinal shear and normal stress in fiber direction using maximum stress theory ..................... 80

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11 LIST OF ABBREVIATIONS ADMM Analytical Direct Micromechanics Method DMM FEA PBC RVE Direct Micromechanics Method Finite Element Analysis Periodic boundary Conditions Representative Volume Element

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12 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science ANALYTICAL AND FINITE ELEMENT BASED MICROMECHANICS FOR FAILURE THEORY OF COMPOSITES By Sai Tharun Kotikalapudi December 2017 Chair: Bhavani V. Sankar Major: Mechanical Engineering An analytical method using elasticity equations to predict the failure of a unidirectional fibe r reinforced composite under multiaxial stress is presented. This technique of calculating micro stresses using elasticity equations and estimating the strengths of a composite is based on the Direct Micromechanics Method (DMM). Prediction of failure using phenomenological failure criteria such as Maximum Stress, Maximum Strain and Tsai Hill theories have been prevalent in the industry. However, DMM has not been used in practice due to its prohibitive computational effort such as the finite element analysis (FEA). The present method replaces the FEA in DMM by analytical methods, thus drastically reducing the computational effort. A micromechanical analysis of unidirectional fiber reinforced composites is performed using the three phase model. A given state of macro stress is applied to the composite and the micro stresses in the fiber and matrix phases and along the fiber matrix inte rface are calculated. The micro stresses in conjunction with failure theories for the con stituent phases are used to determine the integrity of the composite. The analytical model is first verifi ed by comparing with results from finite element based micro mechanics. Then, it is used to study the failure envelopes of various composites.

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13 The eff ects of fiber matrix interface on the strength of the composite is studied. The results are compared with those available in the literature. It is found that the present analytical Direct Micro Mechanics (ADMM) predicts the strength of composites reasonabl y well.

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14 CHAPTER 1 INTRODUCTION Literature Review With the growing application of fiber composites, and with the tremendous progress in low cost manufacturing of composite structures, e.g., wind turbine blades, there is a need to develop efficient predictiv e methodologies for the behavior of composites. This should include probabilistic methods to aid in nondeterministic optimization tools used in design. While methods to predict stiffness properties are well established, methods to predict failure and fract ure properties are still evolving. Computational material science is the new field of study which attempts to use modern computational analysis tools to perform multiscale analysis beginning from atomic scale all the way up to structural scale. While large scale computational methods are being advanced, there is always a need for simple and efficient analytical methodologies. This is especially true for strength prediction and failure behavior of composites. Currently available methods for strength predicti on either use numerical simulations such as finite element analysis [1] or very simple methods such as mechanics of materials models (MoM) [2]. The former can be expensive and time consuming, and the latter is only an approximate estimate to be useful in p ractical design applications. The current study is aimed at developing an analytical micromechanics method that is better than MoM models, but still not as complicated as FEA based micromechanics. To this end we use the principles of Direct Micromechanics Method [3] developed in 1990s in conjunction with the classical three phase elasticity model for unidirectional fiber composites [4]

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15 Several failure theories for composite materials are available in the literature. Majority of them use experimental resul ts along with an empirical (phenomenological) approach to plot the failure envelopes. Contemporary failure theories, developed for unidirectional composites such as Maximum Stress Theory, Maximum Strain Theory and Tsai Hill Theory have been thoroughly stud ied and implemented by various researchers and design engineers. Direct micromechanics method (DMM) has a propitious approach to predict failure strengths for an orthotropic composite material. First proposed by Sankar, it has been widely used to analyze v arious phenomenological failure crit eria, e.g., Marrey and Sankar [5], Zhu et al. [6], Stamblewski et al. [7], karkakainen and Sankar DMM encompasses analytical techniques which is an alternative approach for physical testing and experimental procedures. A micromechanical model is subjected to multiple macro stresses which produce micro stresses in each element of the finite element model. The micro stresses are used to devise a failure envelope considering various failure criteria for fiber and matrix suc h as maximum principal stress theory and von Mises criterion. The interface of fiber/matrix played a pivotal role in the failure of envelopes which was also considered in DMM. Interfacial tensile stress and interfacial shear stress in the composite are ver y sensitive properties that depend on various factors. Research S cope In this section, the research procedure followed will be discussed in detail. The RVE for the analytical approach has been modeled as circular fiber surrounded by an annular region of ma trix The description of analytical model is presented in Chapter 2.

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16 Figure 1 1 Depiction of a RVE for the analytical model. The applied macro stresses on the composite are denoted by the C auchy stress tensor Note that throughout this study the principal material coordinate system of the fiber composite will be denoted either by the standard 1 2 3 coordinates or x y z coordinates used in the commercial finite element software ABAQUS. T he two normal stresses and can be decomposed into two cases hydrostatic stress state such that and a biaxial tension/ compression such that The chart in F igure1 2 depicts various analysi s of the stress cases needed to complete the DMM Figure 1 2 Decomposition of macro stresses applied to an RVE of a fiber composite Applied stress Normal stress Axail and hydrostaic Biaxial tension and compresion Shear stress Transverse Shear YZ Longitudinal Shear XZ Longitudinal Shear XY Fiber Matrix Composite

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17 Application of the stress field on the Representative V olume E lement ( RVE) of the composite through individual cases generates macro strains. Every case has distinct analytical equations for calculating the micro stre sses in the fiber and matrix phases. The load factors are calculated based on the type of failure criterion used either maximum stress or some form of quadratic fai lure criterion, e.g. vo n Mises for isotropic materials. Figure 1 3 Macro stresses applied on the unit cell. (similar to will be acting in the 13 plane and will be acting in the 2 3 plane )

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1 8 Figure 1 4 Decomposition of applied st ate of macro stress es into five cases A) Hydrostatic and longitudinal stress ; B) Biaxial tension and compression ; C) Shear in 2 3 plane ; D) Shear in 1 2 plane ; E) Shear in 1 3 plane (A) Case i (B) Case ii (E) Case v (C) Case iii (D) Case iv

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19 Figure 1 5. Schematic depiction of DMM followed to obtain failure envelopes Figure 1 3 shows the six stresses acting on the unit cell of the composite which are div ided into five cases as shown in figure 1 4. For each case the micro stresses are calculated at several locations using stress equations which is explained in detail in chapt er 2. Figu re 1 5 portrays a schematic of the process followed in Direct Micromechanics M ethod (DMM) to obtain the failure envelopes and strengths. Chapter 2 elaborates the stress equations used for micromechanical analysis and validation of using energy me thods. The analytical equations employed are further validated in Chapter 3 through unit strain analysis in finite element analysis software ABAQUS. Chapter 4 consists of the results obtained from the analytical model wherein a thorough study is performed by considering two different materials i.e. isotropic and transversely isotropic. A meticulous comparison on different strengths of composites with present data is included in Chapter 4 A study is performed in Chapter 4 to understand the effect of fiber v olume fraction on the strength properties for few materials. Macro stresses Isolating stresses/formulating stress equations Micro stresses Eigen values/principal stresses Load Factors Failure envelopes and strengths

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20 CHAPTER 2 ANALYTICAL EQUATIONS Introduction to the Three Phase Model In this section, the three phase concentric cylinder composite assemblage model is described. The three phase model proposed by Christensen [8] had been successfully used in the past for predicting the elastic constants of fiber composites, e.g. Flexible resin/glass fiber composite lamina [9]. In the present study we investigate the use of three phase model to predict the strengths of unidirectional fiber composites. Figure 2 1 Three phase model The model shown in Figure 2 1. c onsists of a single cylindrical inclusion (fiber) embedded in a cylindrical annular region of matrix material. The composite cylinder is in turn embedded in infinite medium properties of which are equal to that of the composite material studied. The fiber and the composite are assumed transversely isotropic and the matrix is isotropic. This enables us to use a polar coordinate system for the analysis. Furthermore, the entire assemblage is in a state of generalized plane strain as the strain 2 1 3 r b a Fiber Matrix Composite

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21 must be uniform an d the same in all three phases. Thus, the problem becomes a plane problem. Since we are using the model to calculate the micro stresses in the fiber and matrix for a give macro stress state, the elastic constants of all three phases must be available for t he stress analysis. As a first step the Rule of Mixtures and Halpin Tsai equations are used to estimate the elastic constants of the composite. In each analysis energy equivalence verifies the validity of the input composite elastic constants as explained in subsequent sections. Halpin Tsai Formulation for Composite Properties Halpin Tsai equation is a widely used semi empirical formulation for transverse moduli of unidirectional fiber composites. The general form of Halpin Tsai equations for a p roperty, say is as follows: Where, : Property of the composite : Property of the fiber : Property of the matrix : Curve fitting parameter : Fiber volume fraction The above formula was obtained using curve fitting the result for squ are array of circular fibers It is found that for an excellent fit is obtained for transverse modulus Whereas for shear modulus the value was in excellent

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22 agreement with the Adams and Doner [ 10 ] solution. In both cases a fiber volume fraction of is used. Since the analytical m odel in this paper has a circular fiber in an annular region of matrix, the curve fitting parameter has been adjusted to formulate more accurate predictions of moduli The curve fitting parameter was estimated for the present case by compari ng the applied macro stresses in each case to the volume average of the corresponding micro stresses. The modified values of are: f or f or transverse shear modulus and for longitudinal or axial shear modulus Tabl e 2 1 Comparison of macro stresses with average micro stresses for longitudinal shear stress c ase Fiber Matrix Average stresses =1 1.235 0 0.647 0 1 0 =1 0 1.235 0 0.647 0 1 Table 2 2 Comparison of macro stresses with average micro stresses for normal s tress and in plane shear stress Case Fiber Matrix Average stresses =1 1.612 0 0 0 0.081 0 0 0 1 0 0 0 =1 0.18 1.166 0.07 0 0.274 0.752 0.105 0 0 1 0 0 =1 0.18 0.07 1.166 0 0.274 0.105 0.752 0 0 0 1 0 =1 0 0 0 1.236 0 0 0 0.647 0 0 0 1

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23 The adjusted curve fitting parameter for transverse modulus is 1.16 c onsidering the average stresses in the composite compared satisfactorily with the input non zero stress for each case as depicted in tables 2 1 and 2 2 To summarize, longitudinal modulus of the composite is calculate d from rule of mixtures : Here, : Longitudinal modulus of the composite : Longitudinal modulus of the fiber : Young s modulus of matrix : Fiber volume fraction Transverse modulus longitudinal s hear modulus and in plane s hear modulus are obtained from Halpin Tsai equations. Longitudinal and Hydrostatic Stress Equations In this section, the displacement equations for the cases of longitudinal and hydrostatic stresses are derived. Since w e are using the composite cylinder model, cylindrical coordinate system is used. The above two cases are also axis symmetric. Since the composite cylinder is under plane strain condition and an axisymmetric model is assumed, we have only one non zero displ acement, which is the radial displacement, The displacement equation for all the three phases (fiber, matrix and composite) is given below.

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24 Here, radial displacement in phase and are constants to be determined using various interface and boundary conditions. The radial displacement equation for the fiber p hase is shown below Here, the subscript denotes all the variables are pertai ning to the fiber p hase. One can deduce as the displacement at the center of the fiber is zero. The str ains are derived as follows: The radial displacement equation for the matrix p hase is derived below T he subscript denotes all the variables are pertaining to the matrix p hase. Following procedures as in fiber phase above we get The radial displacements in th e composite phase are shown below Here, the subscript denotes all the variables are pertaining to the composite phase.

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25 Considering a composite unit cell, the strains along the longitudinal direction can be equated as follows : Where, is the longitudinal strain in the fiber dir ection applied to the composite. Continuity Equations Continuity of displacement and radial stresses must be ensured along the fiber/matrix interface and matrix/composite interface. The interface continuity equations, say between surfaces i and j are given below: From the equation of continuity of displacement along the fiber/matrix interface Here, : Radius of the Fiber Phase From the equation of continuity of radial stress along the fiber/matrix interface The constitutive relation for transversely isotropic materials can be written as Hence, the stress continuity Equation ( 2 19 ) takes the form,

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26 For simplicity, we use the following notations Then Equation ( 2 21 ) can be simplified as Similarly, considering continuity of displacement along the matrix/composite interface we get Equating displacements of matrix and composite phases Here, is the radius of matrix phase Similarly, continuity of radial stress along the matrix/composite interface yield Let Then Equation ( 2 29 ) can be simplified as follows

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27 Boundary Conditions At the radial stress at the boundary of the composite From the constitutive relation we get Here, is the hydrostatic stress applied at the boundary of the composite The constants can be solved from Equation s ( 2 18 ) ( 2 26 ) ( 2 32 ) and ( 2 34 ) The hydrostatic stress is the remote stress applied to the composite. The micro stresses in fiber phase and matrix phase for longitudinal and hydrostatic stresses are calculated from the following equations : Longitudinal Shear Stress in the x y plane In this section, the equations for the case of longitudinal shear stress are derived. Since we are using the composite cylinder model, cylindrical coordinate system is used.

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28 The displacement equation for all the three Phases (fiber, matrix, and composite) is given below: Here, , are radial, angular, and axial displacements in the phase respectively. a re constants varying with stresses in the phase Strain Derivations The six strain c omponents [ are derived in the shear model. From the basic strain formulations for cylindrical system we get, and and It is also observed that the transverse shear strain also vanishes: Hence, only two shear strains will exist in the body, which are derived as follows : and

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29 Boundary Conditions Since the displacement at point is finite the constant pertaining to the fiber phase must be zero. At the longitudinal displacement in the composite phase must be finite, hence The longitudinal shear strains in the composite phase at ca n be derived from Equation s ( 2 48 ) and ( 2 4 9 ) a re shown below : The 3X3 rotation matrix for transformation of cylindr ical coordinates to Cartesian coordinates is Since the transverse shear strain is zero the rotation matrix can be simplified to a 2X2 matrix. Transforming the shear stresses from Equation s ( 2 50 ) and ( 2 51 ) into Cartesian form we get It can be observed that the shear strain is the only shear strain presiding in this model and can be equated to (Constant in composite phase). From the shear stress strain relations formula, we deduce the following relation

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30 Here, : Constant in the composite phase : Longitudinal shear modulus of the composite de rived from Halpin Tsai Equation Continuity Equations Continuity of displacement and radial stresses must be ensured along the fiber/matrix interface and matrix/composite interface. The continuity equations are shown below : Here, , are the radial and axial displacements in consecutive phases and are the radial shear strains in the corresponding phases. From Equation ( 2 5 5 ) continuity of displacement along the fiber/matrix interface at Similarly, at From Equation s ( 2 58 ) and ( 2 59 ) we can infer that Let the const ants and be equal to : Ensuring equal axial displacements at the fiber/matrix interface we obtain where, are constants in fiber and matrix phases

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31 From Equation ( 2 57 ) continuity of shear stress along the fiber/matrix interface where, are the longitudinal shear moduli of the fiber phase and matrix phase respectively Similarly, considering axial displacements at the matrix/composite interface we get Here, are constants in matrix phase and composite phase varying with applied stresses Similarly, continuity of shear stress along the matrix/composite interface yi elds : where, are the longitudinal shear moduli of the matrix phase and composite phase respectively The five constants are solved from the following five equations :

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32 The micro stresses in fiber and matrix phases for longitudinal shear XY case are calculated from the following equations : Longitudinal Shear Stress in the x z plane In this section, the displacement equations for the case of longitudinal shear stress are derived. Since we are using the composite cylinder model, cylindrical coordinate system is used. The displacement equation for all the 3 Phases (Fiber, Matrix and Composite) is given below: Here, , are radial, angular, and axial displacements in the phase respectively. are constants varying with stresses in the phase Strain Derivations The 6 fundamental strains [ are derived in the shear model. From the basic strain formulations for cylindrical system we get,

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33 and and It can be observed that the longitudinal strain by the application of a discrete shear stres s in this model is zero. Now the shear strains are calculated, since the shear is applied along the longitudinal direction it is obvious that transverse shear is zero. Apparently due to the nature of shear stress only two shear strains will exist in the body, which are derived as follows : Boundary Conditions Since the displacement at point is finite the constant pertaining to the fiber phase must be zero. At the longitudinal displacement in the composite phase must be finite, hence The longitudinal shear strains in the composite phase at c an be derived from Equation s ( 2 8 1 ) and ( 2 82 ) a re shown below :

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34 The 3X3 rotation matrix for transformation of cylindrical coordinates to Cartesian coordinates is Since the transverse shear strain is zero the rotation matrix can be simplified to a 2X2 matrix. Transforming the shear stresses from Equation s ( 2 83 ) and ( 2 84 ) into Cartes ian form we get It can be observe d that the shear strain is the only shear strain presiding in this model and can be equated to (Constant in c omposite p hase). From basic shear stress formula, we can deduce the following relation : Here, : Constant C in the composite phase : Longitudinal shear modulus of the composite de rived from Halpin Tsai e quation Continuity Equations Continuity of displacement and r adial stresses must be ensured along the fiber/matrix interface and matrix/composite interface. The continuity equations are shown below :

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35 Here, , are the radial and axial displacements in consecutive phases and are the radial shear strains in the corresponding phases. From Equation ( 2 88 ) continuity of displacement along the fiber/matrix interface we can interpret Similarly, at From Equation s ( 2 91 ) and ( 2 92 ) we can infer that For simplicity, the following assumption has been made and will be considered for future derivations and equations Ensuring equal axial displacements about the fiber/matrix interface at we get Here, and are constants in fiber and matrix phases varying with applied s tresses From Equation ( 2 90 ) continuity of radial shear stress along the fiber/matrix interface

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36 Here, are the longitudinal shear moduli of the fiber phase and matrix phase respectively Similarly, considering axial displacements at the matrix/composite interface we get Here, are constants in matrix phase and composite phase varying with applied stresses Similarly, continuity of radial shear stress along the matrix/composite interface yields Here, : Longitudinal shear modulus of the matrix phase : Longitudinal shear modulus of the composite The five constants are solved from the following five equations : The micro stresses in each phase are calculated from the following equations

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37 Biaxial tension / compression in y z plane In this section, the equations for the case of pure shear in the transverse plane are derived. This two dimensional problem can be dealt by assuming a s uitable Airy stress function which is shown below : Since plane strain is assumed the three non zero stresses pertaining to the 2 D problem [ are de rived from the Airy stress function as shown below The radial stress at any point in the model can be derived using the elasticity equation [1 1 ] as follows The tangential stress derivation is shown below The transverse shear in the plane is derived as follows

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38 The radial strain at any point is given by the following equations H ere, is the plane strain modulus and is the plane strain ratio w hich is given by The radial displacement is ca lculated from the Equation ( 2 113 ) as follows Substituting from E quation ( 2 110 ) in the above equation we get The radial displacement at any point on the surface can be obtained by the following equation :

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39 The tangential strain at any point is given by the following equations : Equation ( 2 117 ) can be modified to obtain the angular displac ement gradient with respect to as follows : Substituting and from Equation s ( 2 110 ) and ( 2 109 ) we get the following relations : The obtained angular displacement gradient is as follows : In the above equations are constants specific to each phase and vary with applied stresses. Since the stresses in fiber at are finite are equal to zero. Cont inuity of must be satisfied at (fiber/matrix interface) and (matrix/composite interface) Continuity of radial stress along fiber/matrix interface yields the following equation :

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40 Similarly, incorporating continuity of radial stress along the matrix/composite interface we get Continuity of radial displacement must be ensured along the f iber/matrix interface Similarly, considering the continuity of radial displacement along matrix/composite interface we get the following equation : Continuity of shear stress must be ensured along t he fiber/matrix interface Similarly, considering the continuity of shear stress along matrix/composite interface we get the following equation :

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41 Continuity of tangential displacement gradient must be ensured along the fiber/matrix interface Similarly, considering the continuity of shear stress along matrix/composite interface we get the following equation : Constants and can be found from the above derived continuity equations. The stresses in each phase can be calculated from the below mention ed equations : Fiber Equations

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42 Matrix Equations Composite Equations Substituting in Equation s ( 2 135 ) i.e. c omposite phase we get

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43 Transforming the stresses into Cartesian co ordinates yields It ca n be observed in Equation ( 2 137 ) the shear in the transverse plane is zero and the only non zero stresses are and which are equal but are acting in opposite directions which is equivalent to shear in 45 degrees. Transverse Shear Equation s In this section, the stress equations for the case of transverse shear yz are derived. This two dimensional problem can be dealt by assuming a suitable stress function [12 ] which is shown below Since plane strain is assumed the three fundamental stresses pertaining to a 2 D problem [ are derived from the Airy stress function The radial stress at any point in the model can be derived using the elasticity equation as follows :

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44 The tangential stress derivation is shown below : The transverse shear in the plane is derived as follows : The radial strain at any point is given by the following equations : Here, is the plane strain modulus and The radial displacement is ca lculated from the E quation ( 2 144 ) as follows Substituting from Eq uation ( 2 141 ) in the above equation we get T he radial displacement at any point on the surface can be obtained by the following equation :

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45 The tangential strain at any point is given by the following equations : Equation ( 2 148 ) can be modified to obtain the angular displacement gradient with respect to as follows : Substituting and from Equation s ( 2 140 ) and ( 2 141 ) we get the following relations : The obtained angular displacement gradient is as follows : In the above equations are constants specific to each phase an d vary with applied stresses. Since the stresses in fiber at are finite are zero. Continuity of must be satisfied at (fiber/matrix interface) and (matrix/composite interface)

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46 Continuity of radial stress along fiber/matrix interface yields the following equation : Similarly, incorporating continuity of radial stress along the matrix/composite interface we get Continuity of radial displacement must be ensured along the fiber/matrix interface Similarly, considering the continui ty of radial displacement along the matrix/composite interface we get the fol lowing equation : Continuity of shear stress must be ensured along the fiber/matrix interface

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47 Similarly, co nsidering the continuity of shear stress along matrix/composite interface we get the following equation : Continuity of tangential displacement gradient must be ensured along the fiber/matrix interface Similarly, considering the continuity of shear stress along matrix/composite interface we get the following equation : Constants can be found from the above derived continuity equations. The stresses in each phase are calculated as follows Fiber Equations

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48 Matrix Equations Composite Equations

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49 Substituting in Equation ( 2 166 ) i.e. composite phase we get Transforming the stresses into Cartesian co ordinates yields We get It ca n be observed in Equation ( 2 168 ) the shear in the transverse plane is the only non zero stress present in the plane whereas and are zero. So, the applied transverse shear can be equated to The micro stresses in fiber and matrix f ro m all the five decomposed cases are superimposed. Principal stresses are calculated from the resultant stresses in both the phases. Depending on the failure criterion used failure envelopes for multiaxial stresses can be developed and various strengths can be predicted. A plane strain case along the transverse plane of the composite is considered to validate the analytical model using finite element analysis which will be discussed in Chapter 3.

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50 CHAPTER 3 FINITE ELEMENT ANALYSIS AND COMPARISON Modelling and analysis of Hexagonal RVE In this section, finite element modelling of a rectangular RVE (representative volume element) of a composite with hexagonal unit cell is discussed. The results will be used in section 3.2 for comparing with analytical results. Only a planar hexagonal array model is considered for the current analysis. Figure 3 1. Representative volume element of a hexagonal unit cell The benefit of using a rectangular RVE is that periodic boundary conditions can be applied with least effort in the finite element a nalysis. Although models with rectangular array of fibers is easier to analyze, the hexagonal unit cell is a better approximation of the three phase model used in the present study. It is also a better idealization of unidirectional fiber composites [13] T he commercial finite element software ABAQUS is used in the present study. The coordinate system used in ABAQUS is depicted in figure 3 2. The conventional principal material coordinate system 1 2 3 is designated as y z x in ABAQUS

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51 Figure 3 2. Coordinate system used in ABAQUS and principal coordinate system A 2D deformable planar shell type element is chosen for the micromechanical analysis. Due to symmetry, a quarter model was considered for the analysis. The section sketch is depicted in the f igure 3 3 For a fiber volume ratio of 0.6, the dimensions of the rectangle are calculated as 15X8. 66 Figure 3 3 Sectional view and dimensions of the RVE Multiple sections were created to obtain a uniform and regular mesh in both the fiber and ma trix phases. Fiber and matrix materials were assumed to be isotropic. The following table 3 1 provides the list of properties used in the FEA.

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52 Table 3 1 Properties of Kevlar/Epoxy used in the FEA Property Fiber Matrix 130 GPa 3.5 GPa 0.3 0.35 Tensile strength 2.8 GP a 0.07 GP a The meshing was fashioned differently for the f iber and matrix phases. Figure 3 4 highlights the different regions as well as the meshing pattern. The region enclosed in a red border is the fiber and rest of the region is matrix. A 4 node bilinear plane strain quadrilateral element ( CPE4R ) was chosen as an element type for both the phases. More information regarding meshing can be observed in the fig ure 3 5. Reduced integration was selected to reduce the computational time. A total of 756 elements were created of which 144 belong to the matrix and 612 belong to the fiber. All the elements in the fiber and matrix phase are quadrilaterals. Figure 3 4 Meshed RVE, red bounded regions represent fiber and green unbounded region represents matrix

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53 Figure 3 5 Element type used for meshing and analysis Linear analysis was performed and micro stresses were calculated at the centroid of each element. From the stress matrix, t he maximum principal stress and von Misses stresses were calculated in each element. Apart from the stresses, volume of each element is also extracted to compute the macro stresses. Three different plane strain analysis were performed on the RVE explicit ly considering unit strain elongation in y and z direction as well as unit strain elongation in both y and z directions together. Figure 3 7 depicts a schematic which summarizes the whole procedure of obtaining the coefficients of stiffness matrix and from the unit strain finite element analysis

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54 Figure 3 6. Boundary conditions and loading in unit strain analysis (A) Direction 2 (B) direction 3 Since we are using plane strain analysis coefficients of stiffness matrix and can also be obtained from the data. The coefficient can be found using the Rule of Mixtures where, is the volume fraction of the fib er, which is assumed to be 0.6 for the analysis and individ ualistic stiffness coefficients and can be calculated from the material properties of fiber and matrix. The stiffness matrix can be populated by considering symmetry of [ C ]. l l b (A) (B) b

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55 Figure 3 7 Schematic of the procedure followed to obtain s tiffness matrix Unit strain analysis Extract S11, S22, and S33 at the centroid of each element Multiply the stresses of each element with their respective volumes Summation of the products divided by the total volume gives t he macro stress Analysis along 2 and 3 directions yields 2 nd and 3 rd Columns of the composite stiffness matrix respectively

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56 Table 3 2. Coefficients of stiffness matrix obtained from unit strain analysis Composite Kevlar/Epoxy 1.63 1.63 0.730 0.730 E glass/Epoxy 1.62 1.62 0.680 0.680 After the micromechanical analysis of the hexagonal RVE one can notice that as well as From table 3 2 it is apparent that analysis along one direction of the transverse axis is sufficient to determine all the coefficients of the stiffness matrix due to symmetry. Figure 3 8 depicts deformed RVE from different unit strain analysis, the unit strain difference can be discerned by overlapping the initial and the deformed bodies. (A) Figure 3 8. Initial and deformed hexagonal RVE under unit strain in A) 2 nd Direction B) 3 rd direction C) 2 nd and 3 rd direction

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57 (B) (C) Figure 3 8 Continued

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58 Unit strain analysis in both 2 and 3 directions, i.e. figure 3 8 C is solely done for comparison of maximum stresses with the analytical model and will be discussed later in this chapter. From the mic r o stresses obtained using the FEA unit strain analysis as well as individual constituent material strengths and failure criteria failure envelopes can be plotted in the 2 3 plane. A flowchart explaining the procedures is given in Fig. 3 9. The stiffness matrix for the case of Kevl ar/epoxy composite has been found out to be Considering the tensile strengths mentioned in table 3 1 for fiber and matrix we determine the load factors in each element for maximum stress criterion and a quad ratic failure criterion. Identifying the maximum load factor from among the element load factors will determine the strength of the composite for the given applied stress. Figure 3 10 shows the comparison of maximum principal stress theory and quadratic th eory (von Mises theory ) in the transverse direction. The difference in the transverse strength is attributed to the hexagonal RVE used for unit strain analysis. Figure 3 10 shows the failure envelopes obtained for Kevlar/epoxy from unit strain analysis in 2 and 3 directions. Table 3 3 contains strengths calculated at different stress conditions. The difference in the values of in direction 2 and 3 can be attributed to the shape of the hexagonal RVE. The strengths obtained are in a good agreement with strengths from Zhu, Marrey, and Sankar [15] finite element model.

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59 Figur e 3 9 Schematic of procedure to plot a failure envelope in 2 3 plane Inputs: Stiffness Matrix and Strengths of the materials Calculating compliance matrix and composite properties Apply a given set of macro stresses on the composite and calculate corresponding macro strains Calculating the micro stresses and principal stresses in fiber and matrix Calculating the load factor for each element using the strength of the material Finding out the maximum load factor and dividing it by the applied stress will result in the maximum stress that can be applied

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60 A three letter notation [ 15] has been used to refer the type of failure criterion used for fiber and matrix in this section as well as in chapter 4. The first letter refers to the failure criterion used for fiber and the second letter refers to matrix failure criterion. The last lett er denotes the type of interface failure incorporated and N denotes that the interface was not considered in the micromechanical analysis. The letters M and Q refer to maximum stress theory and quadratic failure theory, respectively. For example, QQN means that quadratic failure criteria were us ed for both fiber and matrix and interface failure was not considered. Figure 3 10 Failure envelopes of Kevlar/Epoxy in transverse direction obtained through unit strain analysis Table 3 3. Transverse strengths at various points for Kevlar/Epoxy (plane s train) Failure Criteria (Direction 2) (Direction 3) MMN 54 38 58.2 28 QQN 71 59 126 34

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61 Comparison with analytical model In this section comparison of results from finite eleme nt model and analytical model are presented. The validation of the analytical model is done by comparing the unit strain analysis results with finite element unit strain analysis. Direct Micro Mechanics method (DMM) is used in both models to obtain the micro stresses which are used to calculate the load factors (inverse of factor of safety) to develop the failure envelopes. ADMM refers to Analytica l Direct Micromechanics Method and FEA refers to Finite Element Analysis. Figure 3 11 shows the comparison of maxi mum principal stress failure envelopes in the transverse plane. As one can notice the envelopes compare fittingly. Figure 3 11 Comparison of analytical and finite element model failure envelopes using maximum stress theory in the transverse plane (2 3 plane) Figure 3 12 portrays the comparison of the failure envelopes considering quadratic failure theory which also compare favorably. Since the fiber is considered isotropic, von Mises criteria is used.

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62 Fi gure 3 12 Comparison of analytical and finite element model failure envelopes using quadratic theory in the transverse plane (2 3 plane) Table 3 4 shows the comparison of strengths at different locations. One can note the values compare satisfactorily for ( ) pure shear considering quadratic failure for fiber and matrix. Tables 3 5 to 3 10 contains detailed comparison of various stresses i.e. maximum principal stress, maximum von Mises stress. Since failure of an element does not cause catastrophic failure of the composite 10 th percentile st ress in each case is also compared. An additional case for is simulated specifically to compare the stresses at various points. In this case the results compared positively. Relatively noticeable difference can be observed for the case of which can be attributed to the shape of the hexagonal RVE as discussed in the previous section.

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63 Table 3 4 Comparison of various transverse str engths for Kevlar/Epoxy (plane s train) Failure Criteria (Direction 2) (Direction 3) FEA DMM FEA DMM FEA DMM FEA DMM MMN 54 53 38 53 58.2 64 28 44 QQN 71 75 59 75 126 137 34 39 Table 3 5 Comparison of m aximum principal stress for Kevlar/Epoxy ( plane s train) Case Fiber Matrix % Difference FEA DMM FEA DMM Fiber Matrix 20.6 20.2 19.4 20.6 2 6.1 25 20.2 25.2 20.5 19 18.5 27 26.4 27 26.4 2 2.2 Table 3 6 Comparison of m aximum von Mises stress for Kevlar/Epoxy (plane s train) Case Fiber Matrix % Difference FEA DMM FEA DMM Fiber Matrix 14.4 13.2 11.3 11.5 8.4 1.4 15.1 13.2 14.4 11.4 12.7 20.5 10.5 10.6 12.3 11.9 0.2 3.3 Table 3 7 Comparison of average of top 10% maximum principal st resses for Kevlar/Epoxy (plane s train) Case Fiber Matrix % Difference FEA DMM FEA DMM Fiber Matrix 19.4 20.2 19.1 20.4 3.8 6.7 22.1 20.2 23.5 20.4 8.8 13.2 Table 3 8 Co mparison of average of top 10% v on Mises st resses for Kevlar/Epoxy (plane s train) Case Fiber Matrix % Difference FEA DMM FEA DMM Fiber Matrix 12.3 13.2 11 11.1 7.1 0.7 14.4 13.2 12.8 11.1 8.5 13.0

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64 Table 3 9 Comparison of 10 th percentile maximum principal stress for Kevlar/Epoxy (plane s train) Case Fiber Matrix % Difference FEA DMM FEA DMM Fiber Matrix 19.2 20.2 18.9 20.2 5.2 6.9 21.2 20.2 22.8 20.2 4.7 11.4 26.6 26.0 26.6 26.0 2.3 2.3 Table 3 10 Comparison of 10 th percentile maximum von Mises stress for Kevlar/Epoxy (plane s train) Case Fiber Matrix % Difference FEA DMM FEA DMM Fiber Matrix 11.9 13.2 10.8 10.8 10.9 0 13.9 13.2 12.0 10.8 5 10 10.4 10.5 12.2 12.0 1 1.6 To summarize, reasonable comparison of the two models validates the analytical model for further analysis and prediction of composite strengths currently used in indus try which will be discussed in Chapter 4

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65 CHAPTER 4 ANALYTICAL MODEL RESULTS AND DISCUSSION In this section the results obtained from Analytical Direct Micro mechanics (ADMM) will be discussed. The results are divided into four parts: (1) Failure envelopes for composites with isotropic fibers, e.g., Kevlar/epoxy composite; (2) Failure envelopes for comp osites with trans versely isotropic fibers, e.g., carbon/epoxy composite; (3) I nterface effect s on the above composites; (4) V olume fraction analysis for the above mentioned composites; and (5) Strengths for several composites used currently in industries. Results for Kevlar/Epoxy Two failure criteria maximum principal stress and quadratic criteria have been compared in this chapter. The maximum principal stress theory is shown below Where, are stresses in the principal material directions. Strengths and have been tabulated at the e nd of each analysis. Refers to the shear strength in the 2 3 plane. Longitudinal compressive strength has not been studied since the failure due to compressive stress is due to buckling of the fibers and is considered more of an instability phenomenon rather than failure of the material. The notations such as MMN, QQY are defined as follows. The letters M and Q refer to maximum stress theory and quadratic failure theory, respectively. For example, QQN means that quadratic failure criteria was us ed for both fiber and matrix and interface failure was not considered.

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66 In this section, the effective stre ngth properties of Kevlar/Epoxy will be studied using analytical based micromechanics. The fiber and matr ix materials were assumed to be isotropic. Table 4 1 lists the constituent properties used Table 4 1 Properties of Kevlar/Epoxy Property Fiber Matrix 130 GPa 3.5 GPa 0.3 0.35 Tensile strength 2.8 Gpa 0. 0 7 Gpa Figure 4 1. Comparison of MMN and QQN failure envelopes of Kevlar/Epoxy on plane

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67 Figure 4 2. Comparison of MMN and QQN failure env elopes of Kevlar/Epoxy in the plane Figure 4 3. Comparison of MMN and QQN failure e nvelopes of Kevlar/ Epoxy for longitudinal shear in the 1 2 or 1 3 plane

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68 Figure 4 4. Comparison of MMN and QQN failure e nvelopes of Kevlar/Epoxy subjected to both longitudinal and transverse shear stresses Figure 4 5. Comparison of MMN and QQN failure e nvelopes of Kevlar/Epoxy for shear in longitudinal direction and stress in fiber direction

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69 Figures 4 1 through 4 5 shows the failure envelopes of Kevlar/Epoxy for various combinations of normal and shear stresses. The strength properties of Kevlar/epo xy have been extracted from the above plots and are tabulated in table 4 1. The strengths obtained show good comparison with the properties obtained through finite element micromechani cs by Zhu, Marrey and Sankar [16 ]. Table 4 2 Strengths at various poi nts for Kevlar/Epoxy (MPa) Failure Criteria MMN 1573 53 56 44 QQN 1592 75 32 39 Carbon/Epoxy plots In this section, the effective strength properties of carbon/epoxy will be studied using analytical micromechanics. A transversely isotropic fiber is considered for the current analysis. Table 4 3 lists the properties used in detail. Since a transversely isotropic fiber failure is governed generally by Hashin type failure theories [17 ] quadratic failure criterion for tra nsversely isotropic composites is incorporated for carbon with few modifications. The quadratic failure can be summarized into four cases as shown below: Tensile fiber mode (4 2 ) Compre ssive fiber mode (4 3 ) Tensile matrix mode

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70 (4 1 ) Compressive matrix mode (4 2 ) Here refers to the transverse shear strength in the composite. Table 4 3. Properties of Carbon T300/Epoxy 5208 Property Fiber Carbon T300 Matrix Epoxy 5208 Tensile strength Figure 4 6. Comparison of MMN and QQN failure enve lopes of Carbon/Epoxy in 1 2 plane

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71 Figure 4 7. Comparison of MMN and QQN failure enve lopes of Carbon/Epoxy in 2 3 plane Figure 4 8. Comparison of MMN and QQN failure envelopes o f Carbon/Epoxy for shear in longitudinal direction s

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72 Figure 4 9. Comparison of MMN and QQN failure e nvelopes of Carbon/Epoxy subjected to both longitudinal and transverse shear stresses Figure 4 10. Comparison of MMN and QQN failure e nvelopes of Carbon/Epoxy for longitudinal shear and normal stress in fiber direction Figures 4 6 to 4 10 shows the fa ilure envelopes of Carbon/Epoxy for various combinations of normal and shear stresses. The strength properties of Carbon/epoxy

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73 have been extracted from the above plots and are tabulated in table 4 4. The strengths obtained showed excellent comparison with the properties from principles of composite material mechanics by Gibson [ 18 ]. Table 4 4. Predicted strengths of T300/5208/Carbon/Epoxy Failure Criteria MMN 1456 49 57 43 QQN 1456 62 33 31 Effects of Interface In this section effects of interface on different strength properties and envelopes will be discussed. Fiber/ matrix interface plays a pivotal role in determining the strength of the composites. Two failure criteria are used to study the effects of interface on s trengths. It is assumed that compressive stresses will not affect the interface. Maximum stress criteria used is discussed as follows: The normal stress acting at the interface is i.e. radial tensile stress acting at the interface. Interfacial tensile ( Interfacial shear Where is interfacial tensile strength and is in terfacial shear strength

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74 The quadratic interface theory is explained below: Interface strength determination through experimen ts is inaccurate and difficul t as very limited data is available. Significant amount of work was done by Huges [ 19 ] on carbon/Epoxy properties. Many factors were considered and th ere was no simple conclusion found about the nature of Carbon/Epoxy interface. It was found out that the i nterface has a very high stress concertation. The following plots demonstrates the effect of interface on failure envelopes. Figure 4 11. Interface effects on failure envelopes for K evlar/Epoxy in 1 2 plane using maximum stress theory

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75 Figure 4 12. Interface effects on failure envelopes for K evlar/Epoxy in 1 2 plane using quadratic theory Figure 4 13. Interface effects on failure envelopes for K evlar/Epoxy in 2 3 plane using maximum stress theory

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76 Figure 4 14. Interface effects on failure envel opes for K evlar/Epoxy in 2 3 plane using quadratic theory Figure 4 15. Interface effects on failure envelopes subjected to both longitudinal and transverse shear stresses using maximum stress theory

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77 Figure 4 16. Interface effects on failure envelope s subjected to both longitudinal and transverse shear stresses using quadratic theory Figure 4 17. Interface effects on failure envelopes for Ca rbon/Epoxy in 1 2 plane using maximum stress theory

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78 Figure 4 18. Interface effects on failure envelopes f or Ca rbon/Epoxy in 1 2 plane using quadratic theory Figure 4 19. Interface effects on failure envelopes for Ca rbon/Epoxy in 2 3 plane using maximum stress theory

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79 Figure 4 20. Interface effects on failure envelopes for Ca rbon/Epoxy in 2 3 plane using quadratic theory Figure 4 21. Interface effects on failure envelopes for C arbon/Epoxy for envelopes subjected to both longitudinal and transverse shear stresses using maximum stress theory

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80 Figure 4 22. Interface effects on failure envelopes for C ar bon/Epoxy longitudinal shear and normal stress in fiber direction using maximum stress theory Figures 4 11 to 4 22 shows the failure envelopes of Carbon/Epoxy for various combinations of normal and shear stresses. The strength properties of Kevlar/Epoxy a nd Carbon/epoxy have been extracted from the above plots and are tabulated in tables 4 5 and 4 6. The strengths obtained showed acceptable comparison with the properties obtained through finite element micromechanical analys is by Zhu, Marrey and Sankar [20 ]. Table 4 5. Comparison of strengths for Kevlar/epoxy including interface failure obtained using ADMM Failure Criteria MMN 1573 53 56 44 QQN 1592 75 32 39 MMM 1573 23.4 22.55 20.38 QQQ 1592 23.4 18.15 20.38

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81 Table 4 6. Comparison of strengths for Carbon/Epoxy including interface failure obtained using ADMM Failure Criteria MMN 1456 49 57 43 QQN 1456 62 33 31 MMM 1456 38.6 36 36.5 QQQ 1456 39 37 37 Volume fraction analysis In this section the variation of strengths with volume fraction will be discussed for composites Kevlar/Epoxy and Carbon /Epoxy. Figures 4 23 to 4 28 depicts the variation of with Interface failure strengths are also considere d and as explained in Chapter 3 MMM refers to maximum stress theory for fiber, matrix and interface. Figure 4 23. Variation of longitudinal shear s trength with volume fraction in Kevlar/Epoxy

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82 Figure 4 24. Variation of Transverse tensile strength with volume fraction in Kevlar/Epoxy Figure 4 25. Variation of Transverse shear s trength with volume fraction in Kevlar/Epoxy

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83 Figure 4 26. Variation of longitudinal shear s trength with volume fraction in Carbon/Epoxy Figure 4 27. Variation of transverse tens ile s trength with volume fraction in Carbon/Epoxy

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84 Figure 4 28. Variation of transverse shear s trength with volume fraction in Carbon/Epoxy Summary In this section a detailed comparison of strengths obtained from the analytical model with reference str engths extracted from Gibson [21 ] will be discussed. Table 4 7 shows the comparison of properties obtained from maximum principal stress theory and quadratic failure theory with the properties available in literature. von Mises theory was used for an ] criteria was used for transversely isotropic fibers. Table 4 8 shows the percentage difference with reference strengths. It can be noticed that the model showed an excellent comparison with in both MMN and QQN excep t for boron/aluminum and E glass/epoxy. In the case of boron/aluminum yielding plays a crucial role in determining the strengths which the present model will not consider.

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85 Table 4 7. Comparison of strengths for several composites with analytical model st rengths Reference MMN QQN Composite Boron/5505 Boron/epoxy 0.5 1586 62.7 82.7 1593 46 65 1593 63 38 AS/3501 Carbon/epoxy 0.6 1448 48.3 62.1 1456 49 57 1456 62 33 T300/5208 Carbon/Epoxy 0.6 1488 44.8 62.1 1456 48 57 1456 62 33 IM7/8551 7 Carbon/Epoxy 0.6 2578 75.8 --2492 67 69 1492 86 42 AS4/APC2 carbon/PEEK 0.58 2060 78 157 2169 63 68 2169 81 41 B4/6061 Boron/Aluminum 0.5 1373 118 128 400 96 95 400 115 57 Kevalr49/epoxy aramid/epoxy 0.6 1379 27.6 60 1332 38 50 1332 45 31 Scocthply1002 E glass/epoxy 0.45 1100 27.6 82.7 440 35 36 440 44 23 E glass/470 36 E glass/Vinyl ester 0.3 584 43 64 478.5 43 40 478.5 50 24 T able 4 8. %Difference of strengths for several composites relative to reference strengths MMN QQN Composite Boron/5505 Boron/epoxy 0.5 0 1 21 0 31 54 AS/3501 Carbon/epoxy 0.6 0 1 8 0 28 47 T300/5208 Carbon/Epoxy 0.6 2 7 8 2 38 47 IM7/8551 7 Carbon/Epoxy 0.6 3 10 --3 13 --AS4/APC2 carbon/PEEK 0.58 5 18 57 5 5 74 B4/6061 Boron/Aluminum 0.5 67 9 26 67 7 55 Kevalr49/epoxy aramid/epoxy 0.6 3 38 17 3 63 48 Scothply1002 E glass/epoxy 0.45 60 37 56 60 59 72 E glass/470 36 E glass/Vinyl ester 0.3 18 0 38 18 16 63

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86 c omparison has been ac ceptable for all the composites and maximum principal stress theory is found out to be more suitable to calculate transverse tensile strength. On the other hand, there is a lot of discrepancy for but maximum stress theory proved to be best considering the cases of carbon fibers. Predicted strengths for carbon IM7/epoxy have also been presented in table 4 7. Overall maximum principal stress theory overlooked quadratic theories for predicting strengths through Direct M icromechanics Method (DMM).

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87 CHAPTER 5 CONCLUSIONS AND FUTURE WORK The three phase composite model for unidirectional composites which is original ly proposed for homogenization is extended for a method called Analytical Direct Micro Mechanics (ADMM) to predict the strength properties of composites. In the three phase model the fiber is surrounded by an annular region of matrix and the fiber matrix assemblage is embedded in an in finite medium of composite. The elastic constants of the composite are evaluated using a modified Halpin Tsai type equations. In the ADMM the micro stresses are calculated in the fiber and matrix phases for a given macro stress applied to the composite. Th e micro stresses in conjunction with failure theories for fiber, matrix, and fibe r/matrix interface are used to determine the failure of the composites. The ADMM is evaluated by comparing the results with that of finite element based micromechanics. The A DMM results compare reasonably well with FEA based DMM results for failure envelopes. Then the ADMM is extended to various composite systems and compared with available results for strength values. The ADMM is able to predict the strength reasonably well i n majority of the cases. The strength in the fiber direction compares well. The transverse strength pr operties are different in the 2 and 3 direction s in the FEA model because of the hexagonal unit cell. However, the three phase model being axisymmetric pr edicts the same strength in the 2 and 3 directions, which is closer to practica l fiber composites. Since it is based on elasticity solution, it is much faster than FE A based micro mechanics. Due to its speed ADMM can be used in understanding the effects o f variability of constituent properties on the composite

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88 strength. The model will be very much suited for non deterministic design of composite structures. Although present failure theories have been partially successful, many theories have to be modified to keep up with the growing composite technology. The need for unprecedented methods for predicting failure strengths is everlasting. The normal strengths obtained from Direct Micromechanics Method (DMM) were reasonable and are in good agreement with the properties of the composites currently used in the industry. The modified Halpin Tsai formulations played a productive role in calculating the effective composite properties for the three phase model. Although there were slight discrepancies while compari ng ADMM and FEA in principal direction 2 and 3, it still validates the analytical model since symmetry is not observed in a hexagonal RVE. The unit strain finite element analysis of a hexagonal RVE validated the analytical model. Interface strength has no effect on longitudinal tensile strength. Future work The newly developed Analytical Direct Micromechanics Method (ADMM) has a lot of potential for future research. Since a preliminary model was constructed, modifications can be done to consider buck ling of fibers under compression. The ADMM model with correction response surfaces can be a useful analytical tool wherein a polynomial function of design variables is used to rectify the difference from the experimental strength. It is also a menable to pr obabilistic micromechanics, in which the e ffect of variability in fiber and matrix properties on the stiffness and strength properties can be studied. The model can be e xtend ed to fracture analysis (crack propagation in matrix and interface) Since there w as a relatively noticeable difference in longitudinal shear with properties available in literature, more

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89 modifications to the model should be made. Thus, the ADMM paves the way for detailed analysis of the interface.

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90 LIST OF REFERENCES [1] Nam Ho Kim., Co urse material for EML5526 Finite Element Analysis, University of Florida, Gainesville. [2] Abhandlungen der KniglichenGesellschaft der Wissenschaften, Gttingen, Vol. 34, 1887, pp. 3 51. [3] Zhu, H., B. V. Sankar, and R. V. Marrey. "Evaluation of failure criteria for fiber composites using finite element micromechanics." Journal of composite materials 32.8 (1998): 766 782 [4] properties in three phase sphere [5] Marrey, Ramesh V., and Bhavani V. Sankar. "Micromechanical models for textile structural composites." (1995) [6] Stamblewski, Christopher, Bhavani V. Sankar, and Dan Zenkert. "Analysis of t hree dimensional quadratic failure criteria for thick composites using the direct micromechanics method." Journal of composite materials 42.7 (2008): 635 654 [7] Karkkainen, Ryan L., Bhavani V. Sankar, and Jerome T. Tzeng. "Strength prediction of multi la yer plain weave textile composites using the direct micromechanics method." Composites Part B: Engineering 38.7 (2007): 924 932 [8] [9] [10] Journal of Composite Materials Vol. 1, 1967, pp. 152 164. [ 11] Timoshenko.S [12] [13] Agarwal.B. D John Wiley & Sons, Inc (1990) [14 ] Zhu, H., B. V. Sankar, and R. V. Marrey. "Evaluation of failure criteria for fiber composites using finite element micromechanics." Journal of composite materials 32.8 (1998): 766 782

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91 [15] Zhu, H., B. V. Sankar, and R. V. Marrey. "Evaluation of failure criteria for fiber composites using finite element micromechanics." Journal of composite materials 32.8 (1998): 766 782 [16] Zhu, H., B. V. Sankar, and R. V. Marrey. "Evaluation of failure criteria for fiber composites us ing finite element micromechanics." Journal of composite materials 32.8 (1998): 766 782 [17] Hashin, Zvi. "Failure criteria for unidirectional fiber composites." Journal of applied mechanics 47.2 (1980): 329 334 [18] Gibson, Ronald F. Principles of compo site material mechanics. CRC Press, 2011 [19] [20] Zhu, H., B. V. Sankar, and R. V. Marrey. "Evaluation of failure criteria for fiber composites using finite element micromechanics." Jo urnal of composite materials 32.8 (1998): 766 782 [21] Gibson, Ronald F. Principles of composite material mechanics. CRC Press, 2011 [22] Hashin, Zvi. "Failure criteria for unidirectional fiber composites." Journal of applied mechanics 47.2 (1980): 329 334

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92 BIOGRAPHICAL SKETCH Sai Tharun Kotikalapudi was born in Hyderabad, India, in 1991 He grew up in various southern states of India. He received a Bachelor of Technology in the field of mechanical engineering from SASTRA University, India. From there he proceeded to work for a year at Tata Consultancy Services Engineering and Infrastructure Services, in Bangalore, India. He has worked under the guidance of personnel from various internationally recognized companies such as Johnson & Johnso n and Beckman Coulter. His work involved developing and redesigning medical equipment in SolidWorks. He defended his M.S. thesis in October 2017.